LAPLACE. 481 



which as he says is analogous to the corresponding theorem in 

 the Integral Calculus given in Art. 895 ; and, as in that Article, he 

 shews that in the problem he is discussing the exact result lies 

 between two approximate results. See also Art. 770. 



898. The memoir contains on page 287 a brief indication of a 

 problem which is elaborately treated in pages 369 — 376 of the 

 TJi eo rie . . . cles P) 'ob. 



899. Laplace now developes another form of his method of 

 approximation to the value of definite integrals. Suppose we 



require lydx; let Y be the maximum value of ?/ within the 



range of the integration. Assume ?/ = Ye~^\ and thus change 



li/dx into an integi'al with respect to t. The investigation is 



reproduced in the Theor{e...des Frob. pages 101 — 103. 



n 00 



Laplace determines the value of / e~^'dt. He does this by 



taking the double integral / e'^^^-^^'^dsdu, and equating the 



results which are obtained by considering the integrations in 

 different orders. 



900. Laplace also considers the case in which instead of as- 

 suming y = Fe"*^^, we may assume y = Ye~^. Something similar is 

 given in the TJieorie...des Proh. pages 93 — 95. 



Some formulae occur in the memoir which are not reproduced 

 in the Theorie...des Proh., and which are quite wrong: we will 

 point out the error. Laplace says on pages 298, 299 of the 

 memoir : 



, , , , . -iff d.c dz 



Considerons presentement la double integrale 1 1 ■- —3, prise 



;y (1 -z -x^'Y 



depuis a: = jusqu'a £c = l, et depiiis z = ^ jusqu'a ^=1; en faisant 



X , „ , , . [ dz [ dx 



,=x, elle se cnano;era dans ce.le-ci /-— / — , ces 



(l-z')^ Jj{l-z')J{l-x")i 



integrales etant prises depuis x' -0 et ;^ - 0, ju^-qu'a x' —1 et z=l, 



31 



